US 7907998 B2 Abstract A method of determining an impedance of at least one lung of a patient having an implanted pacemaker that comprises first and second leads and a case, the method comprising:
Claims(21) 1. A method of determining a specific impedance of tissue of at least one of two lungs of a patient having an implanted pacemaker that comprises first and second leads and a case, the method comprising:
a) using the first and second leads and the case to acquire at least a first and a second impedance measurement responsive to impedance of the patient's chest, said first and second impedance measurements being at least partially independent; and
b) determining the specific impedance of tissue of at least a first one of the patient's lungs substantially independently of specific impedance of tissue of the second one of the patient's lungs, such that Poisson's equation for a model of the chest has a solution consistent with the at least two impedance measurements.
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a) making an initial guess for values of the specific impedances;
b) calculating expected impedance measurements for said values of the specific impedances;
c) correcting values of the specific impedances according to a difference between measured and expected values for the impedance measurements; and
d) repeating (b) and (c) until a convergence criterion is satisfied.
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19. A method of determining a specific impedance of tissue of at least one lung in a chest of a patient, comprising:
a) generating one or more currents, each between a pacemaker case implanted in the chest, and at least one of a first lead and a second lead implanted at different locations in a heart of the patient;
b) measuring one or more voltages, each involving a potential difference between at least one of the first and second leads and the case, including at least a first voltage due at least in part to at least a portion of the current which is generated between the other one of the first and second leads and the case; and
c) calculating the specific impedance of tissue of at least one lung only from the one or more currents and the one or more voltages, including at least the first voltage.
20. A pacemaker for bio-impedance measurements, comprising:
a) a case adapted for implanting in the chest;
b) a first lead adapted for implanting at a location in the heart;
c) a second lead adapted for implanting at a different location in the heart;
d) a bio-impedance system, adapted to make impedance measurements comprising at least a first and a second impedance measurement, using voltage differences between the case and the first and second leads; and
e) a data analyzer, adapted to determine a specific impedance of tissue of at least a first one of the lungs, substantially independently of specific impedance of tissue of the other lung, such that Poisson's equation for a model of the chest has a solution consistent with the at least two impedance measurements.
21. A method of determining specific impedances of tissue of at least three portions of two lungs of a patient having an implanted pacemaker that comprises first and second leads and a case, the method comprising:
a) using the first and second leads and the case to acquire at least a first, a second and a third impedance measurement responsive to impedance of the patient's chest that are at least partially independent; and
b) using the impedance measurements to determine the specific impedance of tissue of at least three portions of the lungs of the patient, including at least one portion of one of the patient's lungs, and at least two portions of the other one of the patient's lungs, substantially independently of each other.
Description This application is a continuation-in-part of U.S. application Ser. No. 11/023,912 (publication No. 2005/0107719-A1), filed on Dec. 27, 2004, which is a continuation-in-part of U.S. application Ser. No. 10/189,161 (publication No. 2004/0006279-A 1), filed on Jul. 3, 2002. The field of the invention relates to bio-impedance measurements of the chest, particularly for monitoring the lungs and heart of a patient. Patients suffering from CHF, and other conditions that cause pulmonary edema, are typically treated with diuretic drugs to reduce the fluid in their lungs. But these drugs can also cause kidney damage and other medical problems, so it is generally advantageous, in treating these patients, to monitor the level of pulmonary edema, in order to maintain an optimized treatment regime. Levels of pulmonary edema may be designated by a scale that ranges from the least severe cases to the most severe. Bio-impedance techniques offer a non-invasive and inexpensive method for monitoring fluid volume in the lungs. Systems for bio-impedance monitoring of pulmonary edema, developed in the 1970s, are described by R. N. McCarthy and D. Dosh, “Assessment of pulmonary edema in acute congestive heart failure with impedance cardiography,” J. Am. Osteo. Assoc. 74, 879 (1975), and by M. R. Khan et al, “Quantitative electrical-impedance plethysomography for pulmonary oedema,” Med. Bio. Eng. Comp. 15, 627-633 (1977). These systems used two or four electrodes placed on the outside of the body at the upper and lower ends of the thorax. A commercially available system similar to these, the BioZ system sold by Cardiodynamics, is described by B. H. Greenberg et al, “Reproducibility of impedance cardiography hemodynamic measures in clinically stable heart failure patients,” CHF 6, 19-26 (2000). The systems described in the above referenced documents generally yield only an estimate of the average overall thorax impedance, which can be influenced by factors, for example the thickness of surrounding fat layers, other than pulmonary edema. Another kind of bio-impedance system for lung monitoring is usable only for patients who have implanted pacemakers or defibrillators having electrical leads permanently placed inside the heart, typically in the right atrium and the right ventricle. One such system, the Insync Sentry, made by Medtronic, is described by L. Wang et al, “Prediction of CHF hospitalization by ambulatory intrathoracic impedance measurement in CHF patients is feasible using pacemaker or ICD lead systems,” PACE 26, 959 (2003). In this system, current is injected between the right ventricle lead and the case of the pacemaker or defibrillator, and the impedance between these two points is measured. This impedance is found to be well correlated with the overall amount of fluid in the lungs, and is insensitive to external factors such as the thickness of fat layers surrounding the thorax. A. Belalcazar and R. P. Patterson, “Improved lung edema monitoring with coronary vein pacing leads: a simulation study,” Physiol. Meas. 25, 475-487 (2004), describes a bio-impedance system using a new type of implanted pacemaker or defibrillator. The new pacemaker or defibrillator has a lead in the left ventricle coronary vein for improved pacemaking or defibrillating functioning, in addition to leads in the right atrium and right ventricle. Like many pacemakers and defibrillators, the described pacemaker or defibrillator has its case implanted in the left pre-pectoral position. Measuring impedance between each of these leads and the case allows changes in impedance due to pulmonary edema to be evaluated independently of changes in impedance due to intravascular fluid. The left ventricle coronary vein lead is said by Belalcazar and Patterson to be important for monitoring edema, because impedance measurements using this lead are more sensitive to pulmonary edema, and less sensitive to the volume of intravascular fluid, than impedance measurements using the right atrium and right ventricle leads. Impedance measurements using any of the leads in the devices described by Belalcazar and Patterson are said to be sensitive only to left lung edema, not to right lung edema, in patients where the degree of edema is very different in the two lungs (unilateral edema). Whereas it is pointed out that for most pulmonary edema patients, fluid is distributed approximately uniformly in both lungs, pulmonary edema concentrated asymmetrically in the right lung does occur and can be a symptom of mitral regurgitation. The authors state that, for detecting pulmonary edema, “The right lung can only be monitored with a right pectoral implant.” Ziochiver, Rosenfeld and Abboud, in “Induced Current Electrical Impedance Tomography: A 2-D Theoretical Simulation,” IEEE Transactions on Medical Imaging 22, 1550-1560 (2003), describe an impedance imaging system in which eight or more electrodes are arranged around the outside of the chest. Impedances are measured between different pairs of the electrodes, and the measurements are used to calculate an impedance image of the chest, with sufficiently high resolution to “see” each of the two lungs and the heart. Abboud is also an inventor on published U.S. application 2004/0006279-A1, which describes a similar system, and on the present application. The disclosures of all the patents and publications mentioned above are incorporated herein by reference. An aspect of some embodiments of the invention relates to determining impedance of each of a patient's lungs responsive to at least two impedance measurements, acquired using first and second electrical leads of an implanted pacemaker. In some embodiments of the invention, a defibrillator is used instead of a pacemaker, or a combined unit is used which acts both as a pacemaker and a defibrillator. It should be understood that, whenever a pacemaker is mentioned, a defibrillator or combined unit may be used instead. An aspect of some embodiments of the invention, relates to providing relatively accurate determinations of impedance of a lung of a patient responsive to the at least two impedance measurements. According to an aspect of some embodiments of the invention, the at least two impedance measurements put at least two constraints on solutions to Poisson's equation, in a model of the chest in which the impedances of the two lungs are treated as independent variables. The impedance of a given lung is found for which the solution to Poisson's equation satisfies the constraints. In an embodiment of the invention, current is generated between the first lead and the case, and voltage is measured between the first lead and the case to make a first impedance measurement of the at least two impedance measurements. Voltage is measured between the second lead and the case to make the second impedance measurement. In another embodiment of the invention, current is generated between the second lead and the case to make the second impedance measurement. In other embodiments of the invention, to make one or more of the impedance measurements, current is generated between each of the leads and the case, with a known current passing through each of the leads, or a known relationship between the currents that are passed through the leads. In some embodiments of the invention, to make one or more of the at least two impedance measurements, the measured voltage is a combination, for example a particular weighted average, of the potential differences between each of the leads and the case. By using different configurations of the two leads when making voltage measurements and/or when generating currents, as described above by ways of example, different impedance measurements involve currents that are distributed differently in the chest and/or voltages between different parts of the chest. The different impedance measurements will therefore generally have different dependences on the impedances of the left and right lungs. The different impedance measurements define a set of two independent constraints on a solution to a Poisson equation for the chest. A lung impedance for either of the lungs or both of the lungs is determined responsive to a solution of the equation that satisfies the constraints. The inventors have determined that in general, requiring a solution to Poisson's equation to satisfy at least two independent conditions, corresponding to at least two impedance measurements acquired using electrodes of an implanted device, provides estimates of lung impedance that are more accurate than estimates provided by prior art methods using implanted devices. Optionally, the first and second leads are located in the right atrium and the right ventricle, as in a traditional pacemaker. Optionally, the case is implanted in the traditional left pre-pectoral position. Alternatively, one of the leads is optionally located in the left ventricle coronary vein, and/or the case is implanted in the right pre-pectoral position, as in some new pacemakers. The invention may thus be used in patients implanted with any of a variety of pacemaker types and configurations, rather than being limited to a particular configuration or type. Furthermore, by making use of the different dependences of impedance measurements on left and right lung impedance, the right lung impedance, for example, may be found even if both impedance measurements are more sensitive to the left lung impedance, which is often the situation when a pacemaker is implanted on the left side of the chest. An aspect of an embodiment of the invention relates to determining lung impedance at least in part from an impedance measurement that is a ratio of voltage between one lead of an implanted pacemaker and its case, and a current generated at least partly between the other lead and the case. It should be understood that, rather than making impedance measurements, and solving equations involving the impedance measurements to find the lung impedances, as described above, optionally voltage and current measurements used to determine the impedance measurements are used directly in equations that are solved to find the lung impedances. As used herein, “making impedance measurements” includes measuring any quantities, such as voltage and current, which could be used to determine an impedance measurement, even if the impedance measurement is not explicitly calculated. Pulmonary edema is primarily a condition of excess extracellular fluid in the lungs. For frequencies below 100 kHz, bio-impedance measurements in general depend substantially on impedance of extracellular fluid, which is resistive, and are substantially independent of impedance of intracellular fluid which is substantially reactive. For frequencies below about 100 kHz, intracellular current does not substantially contribute to total current between electrodes used to measure impedance, since for these relatively low frequencies current does not easily cross the cell membranes. Generally, frequencies below 100 kHz are used for purposes of monitoring pulmonary edema, and the lung impedances as well as the measured impedances are predominantly resistive. At higher frequencies, body impedance depends also on intracellular fluid and generally has a substantial reactive part, since the cell membranes act like capacitors and contribute displacement current to total current between electrodes used to measure impedance. For some medical conditions, which involve abnormal levels of intracellular fluid, it may be advantageous to determine impedance in accordance with an embodiment of the invention at frequencies above 100 kHz, and to determine both the resistive and reactive part of the lung impedances. As used herein, “lung impedance” means the specific impedance of the lung, a a bulk property of the lung tissue, which includes the specific resistance (resistivity) of the lung tissue, and may also include a reactive part. Optionally, values for the left and right lung impedances, consistent with the impedance measurements, are found iteratively by first estimating impedances for the left and right lungs. Expected voltages between each of the two leads and the pacemaker or defibrillator case for given currents are then determined responsive to the estimated lung impedances. The expected voltages are compared to the measured voltages at the two leads. Differences between the expected and measured voltages are used to produce new estimates for the left and right lung impedances, using for example a Newton-Raphson method, or a modified Newton-Raphson method. The procedure is iterated until a convergence criterion is satisfied. There is thus provided, according to an exemplary embodiment of the invention, a method of determining an impedance of at least one lung of a patient having an implanted pacemaker that comprises first and second leads and a case, the method comprising:
Optionally, using the measurements to determine the impedance of at least one of the lungs comprises determining the impedance of said lung such that Poisson's equation for a model of the chest has a solution consistent with the at least two impedance measurements. Optionally, the model of the chest treats the impedances of the two lungs as two independent variables. Optionally, acquiring the impedance measurements comprises, for each impedance measurement:
Optionally, for the first and second impedance measurements, the current is generated only between the case and one of the leads, the same lead for both the first and second impedance measurements, and the voltage involves only a potential difference between the case and one of the leads, a different one for the first impedance measurement than for the second impedance measurement. Optionally, determining the impedance of at least one lung comprises determining the impedances of both lungs. Optionally, the impedance measurements also comprise a third impedance measurement. In an embodiment of the invention, the method also includes determining a third parameter of the thorax, in addition to the two lung impedances. Optionally, the third parameter comprises an intravascular volume of the thorax. Alternatively, at least one of the lung impedances pertains only to a portion of said lung, and the third parameter comprises an impedance of another portion of said lung. Optionally, a least squares method is used to find a fit of the two lung impedances to the at least three impedance measurements. In an embodiment of the invention, determining the lung impedances comprises:
Optionally, calculating the expected impedance measurements comprises using one or more of the finite element method, the finite difference method, the finite boundary element method, and the finite volume method. Optionally, one of the leads is implanted in the right ventricle. Optionally, one of the leads is implanted in the right atrium. Optionally, one of the leads is implanted in the left ventricle coronary vein. Optionally, the case is implanted in the left pre-pectoral position. Alternatively, the case is implanted in the right pre-pectoral position. There is further provided, in accordance with an exemplary embodiment of the invention, a method of determining the impedance of at least one lung, comprising:
There is further provided, in accordance with an exemplary embodiment of the invention, a pacemaker for bio-impedance measurements, comprising:
Exemplary embodiments of the invention are described in the following sections with reference to the drawings listed. The drawings are not necessarily to scale and the same reference numbers are generally used for the same or similar features that are shown on different drawings. When pacemaker 102 makes impedance measurements, there are several ways of making the measurements which are conventionally used to determine lung impedance. For example, pacemaker 102 may generate a given current through one of the leads, and measure a resulting voltage on the same or a different one of the leads. Alternatively, pacemaker 102 may apply a given voltage to one of the leads, or given voltages to each of the leads, and measure the resulting currents. Alternatively, pacemaker 102 may apply voltage and current related in a given way, for example a given voltage in series with a given resistance to limit current. Using a configuration that puts limits on both the current and the voltage may be advantageous for safety reasons. In the preceding examples, each impedance measurement is found by dividing a voltage by a current, regardless of whether the voltage or the current or a combination of the two is fixed. Alternatively, pacemaker 102 makes an impedance measurement directly, without first finding a voltage and a current and dividing the voltage by the current. In describing The currents and voltages used for providing bio-impedance measurements are configured so that they substantially do not interfere with vital functioning of the pacemaker. Any of various methods and devices known in the art may be used to appropriately configure the currents. For example, the currents may be configured so that their magnitudes are sufficiently low and/or their frequencies sufficiently high so that they do not interfere with the pacemaker functions. To substantially prevent “pacemaking” currents and voltages generated by the pacemaker from interfering with bio-impedance measurements, the measurements and pacemaker currents and voltages are optionally synchronized so that they are not simultaneous. Optionally, the impedance measurements and pacemaker currents and voltages do not interfere with each other because they are characterized by frequencies that are sufficiently different. Currents typically used for bio-impedance monitoring are less than about 10 mA, and are characterized by frequencies in a frequency range between 20 kHz and 200 kHz. To measure excess fluid in the lungs associated with pulmonary edema, frequencies below 100 kHz are typically used, for the reasons explained above. At these lower frequencies, the impedance of the lungs and the rest of the chest is primarily resistive, and depends largely on the volume and distribution of low-resistivity blood and other extracellular fluid in the chest. Other body tissues, such as fat, muscle, and bone, as well as air in the lungs, have higher resistivities than blood and other extracellular fluids in the chest. The relationship between an average impedance and degree of edema of each of the lungs, and voltage measured at each of leads 104 and 106 for given injected currents, may be modeled using suitable methods known in the art. For example, the relationship may be numerically calculated using a model of the spatial distribution and respective specific impedances of different body tissues in the chest, and a variable impedance for each of the lungs. The impedance of the lungs, in turn, may be related to a degree of lung edema, using a mathematical model and/or empirical data. P. Nopp, D. Harris, T. X. Zhao, and B. H. Brown describe a lung model suitable for the practice of the invention in an article titled “Model for the dielectric properties of human lung tissue against frequency and air content,” Medical and Biological Engineering and Computing 35, 695-702 (1997), the disclosure of which is incorporated herein by reference. Current may be generated through leads 104 and 106 sequentially, or simultaneously through both leads. When generated simultaneously, impedance measurements are made for at least two different ratios between the injected currents in order to provide at least two impedance measurements, but in different ratios for making different impedance measurements. Current generated through lead 104 goes through chest 100 along paths 202 (in Current generator 404 sends voltage measurements 406 for each of the two leads, relative to the voltage of case 110, to a data analyzer 408, either directly, or through controller 402. Data analyzer 408 is optionally housed in pacemaker 102, for example as part of controller 402. Alternatively, data analyzer 408 is located outside the body, and optionally communicates with pacemaker 102 by a wireless method of communication. Although the pacemaker could be connected to an external data analyzer by wires, any wires extending from the pacemaker to the outside of the body may increase the risk of infection. Even if data analyzer 408 is housed in pacemaker 102, a communications link to the outside of the body, preferably a wireless link, is optionally used to communicate output of the data analysis, for example estimates of the lung impedances, as well as to receive instructions, for example instructions to initiate impedance measurements. In some embodiments of the invention, controller 402 controls current generator 404 to make at least two separate impedance measurements, generating current and/or measuring voltage using different leads, or using the leads in different ways, for each impedance measurement. Optionally, current generator 404 makes voltage measurements using only one lead for each impedance measurement, for example only the lead through which current is being generated. Alternatively, for at least one of the impedance measurements, current generator 404 also measures the voltage of one of the leads through which current is not being generated. In some embodiments of the invention, current is generated through a same one of the leads for both impedance measurements, and the two impedance measurements are distinguished by using different leads for the two voltage measurements. In some embodiments of the invention, current is simultaneously generated through both leads, for each impedance measurement, but with different ratios of amplitude and/or different relative phases for different impedance measurements. In general, voltage may be measured on a same one of the leads or on a different one of the leads during each impedance measurement. In some embodiments, voltage is measured on both of the leads during each impedance measurement, and a weighted average is taken of the two voltages, optionally using a different weighting for each of the impedance measurements. In some embodiments of the invention, the different impedance measurements are made at different times. Alternatively or additionally, different impedance measurements are distinguished by being made at different frequencies. In some embodiments of the invention, only a single impedance measurement is made, using a voltage on one lead that is due at least partly to a current generated into the other lead. Data analyzer 408 uses the voltage measurements and the values of the generated currents to calculate output 410 relating to pulmonary edema. For example, output 410 comprises estimates of the impedance of each lung, and/or an evaluation of the degree of edema in each lung, and/or an evaluation of whether the edema is getting worse in one or both of the lungs. Alternatively, output 410 only includes an estimate of the impedance or degree of edema of one lung, for example if the patient is known to have severe edema only in one lung, which needs to be monitored. However, since the method of calculating the lung impedance makes it easy to calculate the impedance of each lung, and in many patients it is important to monitor both lungs for edema, it is usually advantageous to include estimates for each lung in output 410. Optionally, data analyzer 408 also communicates with controller 402, for example in order to find out from controller 402 what currents were being generated during the time that the voltage measurements were made, or to tell controller 402 what currents to generate. Data analyzer 408 may be a general purpose computer, particularly if data analyzer 408 is located outside the body. Alternatively, data analyzer 408, and optionally controller 402 as well, comprises specialized electronic circuitry adapted for the functions it performs, for example any suitable ASIC or generalized chip. At 502, a model of thorax impedance is created for the patient being examined. The model is, for example, a three-dimensional model describing the boundaries of the different types of tissue in the thorax, including blood, lungs, heart muscle, skeletal muscle, bone, fat, and skin. The boundaries of the different types of tissue are determined, for example, by using a three-dimensional medical image, such as an MRI or CAT scan, of the patient, or by using a standard model of a human thorax based on other subjects, optionally similar to the patient in one or more of height, weight, body build, chest measurement, body mass index (BMI), body surface area (BSA), and sex. A specific impedance is assigned to each tissue, other than the lungs. The assignment of lung impedance is described below. Optionally, the impedances assigned to each tissue are purely resistive. This may be done, for example, if the impedance measurements are to be made at relatively low frequencies, below 100 kHz, because only the resistive parts of the lung impedances are of interest. In general, the specific impedances of body tissues may be frequency dependent, and tissue impedances are assigned that are appropriate for the frequency at which the impedance measurements are to be made. The model optionally assigns a specific impedance, or resistivity, to each volume element (voxel) in a Cartesian grid covering the thorax, or to each element in a more general polyhedral grid, for example a grid in which many grid points are located at the boundaries between different types of tissue, and at the surface of the leads and the pacemaker case. Optionally, the choice of grid used in the model depends on the numerical method that is used to solve the Poisson equation, as will be described below. For example, a Cartesian grid may be chosen if a finite difference method is to be used, while a more general grid may be chosen if a finite element or finite volume method is to be used. At 504, initial guesses are made for the left lung impedance, and the right lung impedance, at the frequency at which impedance measurements are to be made. The initial guesses are based, for example, on typical values of lung impedance for healthy subjects, and/or on values of lung impedance calculated for this patient in an earlier bio-impedance measurement, and/or on typical values of lung impedance for patients with the same degree of pulmonary edema that this patient is believed to have, based, at least in part, on other diagnostic methods. At 506, a Poisson equation, Since the leads and the case are good conductors, the potential has a constant value everywhere on the surface of the case, and has a constant value, in general different from the potential at the case, everywhere on the surface of lead 104, and a third constant value everywhere on the surface of lead 106, generally different from the potentials at the case and at lead 104. Poisson's equation is optionally solved subject to a boundary condition that each of these surfaces is an equipotential. Optionally, another boundary condition is that the current integrated over the surface of each lead is equal to the current generated through that lead. Optionally, another boundary condition is that the current density integrated over the surface of the case is equal to the negative of the current going into the case, which is the total current generated through the leads. Neumann boundary conditions, specifying a normal electric field or zero, are optionally chosen for the vertical and radial surfaces of the thorax model, since no current flows outside the body, and very little current will extend past the vertical limits of the thorax model if they are far enough away from the heart, lungs, and pacemaker. Since exact solutions to the Poisson equation generally do not exist for a complicated geometry such as that of the thorax, numerical methods may be used. Any known numerical method of solving partial differential equations is optionally used, for example the finite difference method, the finite element method, the finite boundary element method, or the finite volume method. These methods generally provide values of the potential ψ at each point on a grid, and various interpolation methods may be used to find ψ at other points. Numerical methods in which arbitrary grid points (rather than Cartesian grid points) can be used may have a potential advantage, since grid points may be chosen to cover the surfaces of the leads and the pacemaker case fairly densely, while fewer grid points are used inside large volumes with uniform resistivity. The solution found at 506 provides a potential ψ_{s }of each lead, relative to the pacemaker case. Optionally, these potentials are found first for a situation of current generated through one lead, and then for a situation of current generated through the other lead, or for two different ratios of current generated through the two leads, if measurements of voltage are made for two different situations of current generation. Thus, up to four independent potentials ψ_{s }may be found, corresponding to voltage measured on each of the two leads, with current generated through each of the two leads. Alternatively, only two or three of these potentials ψ_{s }are found. The independent potentials ψ_{s }found by solving the Poisson equation may be symbolized by a vector {right arrow over (ψ)}_{s}, while the corresponding measured potentials ψ_{0 }are symbolized by a vector {right arrow over (ψ)}_{0}. At 508, the difference vector {right arrow over (ψ)}_{s}−{right arrow over (ψ)}_{0 }between the calculated and measured potentials is found. At 510, the difference vector is examined to see whether it is small enough to satisfy a convergence criterion. The convergence criterion may be satisfied on any of the components of the vector or any combination of the components, but typically the convergence criterion is only satisfied if all of the components are sufficiently small. For example, the convergence criterion requires that the L2 norm of {right arrow over (ψ)}_{s}−{right arrow over (ψ)}_{0 }(the Euclidean distance between {right arrow over (ψ)}_{s }and {right arrow over (ψ)}_{0}) be smaller than a given value, or smaller than a given fraction of the magnitude of {right arrow over (ψ)}_{s}, such as 10^{−2 }of the magnitude, or 10^{−3 }of the magnitude, or 10^{−6 }of the magnitude, or 10^{−10 }of the magnitude. Simulations show that the L2 norm of {right arrow over (ψ)}_{s}−{right arrow over (ψ)}_{0 }typically converges to within about 10^{−10 }of the magnitude of {right arrow over (ψ)}_{s }in about 10 iterations. If the convergence criterion is too stringent, for example less than the round-off error of the computer, it may never be met, or the computation may take an unnecessarily long time. If the convergence criterion is not stringent enough, the output values for lung resistivity may not be very accurate, and the method may not evaluate the degree of pulmonary edema very accurately. For clarity of exposition, we first describe an embodiment of the invention where only two components of {right arrow over (ψ)}_{s }and {right arrow over (ψ)}_{0 }are found, and where each lung impedance is assumed to be uniform, and all other specific impedances in the thorax are assumed to have fixed values. Later we will describe other embodiments. The two lung impedances ρ_{left }and ρ_{right }may be symbolized by a two-component vector {right arrow over (ρ)}_{k}, where k is an iteration number. The vector {right arrow over (ψ)}_{s }of potentials found by solving the Poisson equation depends on the vector of lung impedances {right arrow over (ρ)}_{k}. The vector {right arrow over (ψ)}_{s }may therefore be written as {right arrow over (ψ)}_{s}({right arrow over (ρ)}_{k}). At 512, the Jacobian matrix _{k }is found. This matrix, which is a 2×2 matrix for the embodiment we are considering, is defined by At 514, a new vector {right arrow over (ρ)}_{k+1 }for the lung impedances is found by using the equation
Then the Poisson equation is solved again at 506, as described above, using the new values of lung impedance, and the loop repeats until the convergence criterion is met at 510. Optionally, if the convergence criterion is not satisfied after a certain number of iterations, then the program exits and indicates a failure to converge. When the convergence criterion is met at 510, the final values for lung impedance are output at 516. Alternatively, only one of the values of lung impedance is output at 516, for example if only one of the lungs is being examined for edema, or only an average value for the two lungs is output at 516. In some embodiments of the invention, three or four different potentials are measured and calculated, so _{s }and _{0 }have three or four components instead of only two components. In some such embodiments, the number of variable impedance parameters is equal to the number of potentials, so the vector _{k }of impedances has the same number of components as _{s }and _{0}. For example, it may be useful to include an effective blood resistivity as a varying impedance parameter, together with the left and right lung impedances. Although the resistivity of blood may not really vary significantly, treating it as a free parameter may be used to simulate changes in intravascular volume. Patients with pulmonary edema often also have changes in intravascular volume, which can affect the impedance of the thorax, and may produce misleading conclusions about changes in lung impedance. Including the effective blood resistivity as a free parameter may produce more accurate values of lung impedance for these patients. It may also be useful to divide up one or both lungs into two or more regions, and treat the specific impedance of each region as an independent parameter. Although resistivity tends to be fairly uniform throughout each lung in the early stages of pulmonary edema, more fluid may accumulate near the bottom of the lungs in later stages, and dividing each lung into an upper and lower region, for example, for a total of four components of lung impedance, may produce more accurate evaluations of pulmonary edema in patients with later stage disease.When the number of components of the impedance vector {right arrow over (ρ)}_{k }is equal to the number of components of the potentials {right arrow over (ψ)}_{s }and {right arrow over (ψ)}_{0}, then the iterative method described above for two components may still be used, the only difference being that the vectors have three or four components instead of two components, and the Jacobian matrix is 3×3 or 4×4. In some embodiments of the invention, more components of potential are measured and calculated, than there are free parameters of the impedance. In this case, the problem of calculating the lung impedances from the measured potentials is over-determined, and may be solved using any of the standard methods used for solving over-determined problems, for example the least squares method, described by D. S. Watkins, Fundamentals of Matrix Computations (Wiley-Interscience, 2002), the disclosure of which is incorporated herein by reference. The redundancy provided by the extra potential measurements may result in more accurate values for the lung impedances, and also provides a check on the self-consistency of the data and the model. It should be noted that even if the model used for the thorax does not result in output values for the lung impedances that are very accurate absolutely, the model may still provide an accurate assessment of whether the lung impedances have changed since the last time the patient was tested, and in what direction it has changed. For purposes of treating a patient with pulmonary edema, knowing whether and to what extent the lung impedances have changed may be more important than knowing their values absolutely at a given time. The disclosures of all publications cited here are incorporated herein by reference. The invention has been described in the context of the best mode for carrying it out. It should be understood that not all features shown in the drawings or described in the associated text may be present in an actual device, in accordance with some embodiments of the invention. Furthermore, variations on the method and apparatus shown are included within the scope of the invention, which is limited only by the claims. Also, features of one embodiment may be provided in conjunction with features of a different embodiment of the invention. As used herein, the terms “have”, “include” and “comprise” or their conjugates mean “including but not limited to.” As used in the claims, “pacemaker” also includes defibrillators, and “impedance,” when referring to a lung, means “specific impedance.” Patent Citations
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